1 - Aspects of Differential Geometry

Summary

This Chapter describes, in concise manner, aspects of differential geometry that are necessary to follow the developments of this book. We give several definitions of the concept of the manifold, illustrated by a number of examples. We then define differential forms, which are viewed as the most primitive objects one can put on a manifold. We define their wedge product and the operation of exterior differentiation. We then define the notions necessary to define the integration of differential forms. After this we define vector fields, their Lie bracket, interior product, then tensors. We then describe the Lie derivative. We briefly talk about distributions and their integrability conditions. Define metrics and isometries. Then define Lie groups, discuss their action on manifolds, then define Lie algebras. Describe main Cartan's isomoprhisms. Define fibre bundles and the Ehresmann connections. Define principal bundles and connections in them. Describe the Hopf fibration. Define vector bundles and give some canonical examples of the latter. Describe covariant differentiation. Briefly reivew Riemannian geometry and the affine connection. We end this Chapter with a description of spinors and their relation to differential forms.